Depay, Eric Garcia and Rey Manaj could only be registered by Barca on the eve of Sunday's game at Camp Nou because Pique took a substantial pay cut, ensuring the cash-strapped Catalan club met LaLiga's salary cap limit.
Impressive debutant Depay set up Pique for the opening goal after chants of 'Messi' reverberated around the stadium in the 10th minute of the Blaugrana's first game since his sensational move to Paris Saint-Germain.
Martin Braithwaite was on target at the end of the first half and added a second after the break before late goals from Julen Lobete and Mikel Oyarzabal set up an unlikely tense finale.
Sergi Roberto eased the nerves with a fourth Barca goal in injury time as they started the campaign with a richly deserved victory.
Antoine Griezmann headed against the crossbar as La Real struggled to contend with Barca's crisp passing and movement in a blistering start from Ronald Koeman's side, but Pique opened the scoring after 19 minutes.
Depay had been lively from the start and the Netherlands forward whipped in a pinpoint free-kick for the unmarked centre-back, who headed in from eight yards out.
Brave goalkeeping from Alex Remiro denied Depay a debut goal soon but Braithwaite doubled vibrant Barca's lead in first-half stoppage time, planting a powerful header into the net after Frenkie de Jong stood up an inviting cross.
Jorda Alba failed to hit the target with only Remiro to beat after he was sent clear early in the second half, but he made amends just before the hour mark.
Depay slipped the marauding full-back in with a clever pass and Remiro could only palm his cross into the path of Braithwaite, who fired home from six yards out.
Barca were relentless and Pedri forced a fine reflex save from the busy Remiro before Lobete pulled a goal back with a measured left-foot finish.eight minutes from time and Oyarzabal curled home a sublime free-kick.
Sergi Roberto sealed the points, though, tucking home the excellent Braithwaite's cross from the right.